An analytical approximation of the evolution of the primordial curvature perturbation in the ultraslow-roll inflation: an extended study

Cosmic inflation can enter an ultraslow-roll (USR) stage, if there is a plateau on the inflaton potential. During this stage, the primordial curvature perturbation Rk and its power spectrum PR can be remarkably enhanced on small scales. In this work, an analytical approximation is provided to systematically study the evolution of Rk in the USR inflation. We first discuss the asymptotic solutions of the moduli and arguments of Rk and its time derivative Rk,N on the sub- and super-horizon scales separately and find that all these solutions have simple exponential forms. Then, Rk on five typical scales are investigated in order. Our analytical approximation predicts that Rk first revolves around the origin in the complex plane, but if it crosses the horizon around the start of the USR stage, there appears a subsequent linear evolution towards or away from the origin. This behavior naturally explains the shape of PR from the sharp dip to the peak and matches the numerical results perfectly. Moreover, the minimum of PR is exactly proved to be nonvanishing. Our analytical approximation will help the model building in primordial black hole and gravitational wave physics.